= Lifting-line theory
{wiki}
\b[Lifting-line theory] represents a finite <aircraft wing> by a spanwise <bound vortex> whose <circulation> varies with position. The trailing <vortex filaments> induce <downwash>, reducing the local effective <angle of attack>. In its classical form, the model assumes steady, incompressible <potential flow>, a comparatively large <wing aspect ratio>, small sweep and locally approximately two-dimensional section behavior.<Prandtl's Lifting Line Introduction (16.100) by MIT OpenCourseWare>[p.1, assumptions]{ref}
Let $y$ run along the <wingspan>, let $c(y)$ be the local <airfoil chord>, and take downward <downwash> $w(y)>0$. Combining the <Kutta–Joukowski theorem> with a linear section law gives
$$
L'(y)=\rho U\Gamma(y),\qquad c_l(y)=\frac{2\Gamma(y)}{Uc(y)}=a_0(y)\left[\alpha(y)-\alpha_0(y)-\frac{w(y)}U\right].
$$
Here $a_0$ is the <section lift-curve slope>, $\alpha_0$ the <zero-lift angle of attack>, and $w/U$ the small <induced angle of attack>. Because the wake induced by the entire circulation distribution determines $w(y)$, these equations couple different sections. Solving independent two-dimensional <airfoils> and adding their <lift> misses that coupling.<Prandtl's Lifting Line Introduction (16.100) by MIT OpenCourseWare>[pp.2–3, effective angle and fundamental equation]{ref}
An <elliptic span loading> has $\Gamma(y)=\Gamma_0\sqrt{1-(2y/b)^2}$ and constant $w=\Gamma_0/(2b)$. Integration gives
$$
L=\frac{\pi\rho Ub\Gamma_0}{4},\qquad \alpha_i=\frac{L}{\pi q b^2},\qquad D_i\simeq L\alpha_i=\frac{L^2}{\pi q b^2},\quad q=\frac12\rho U^2.
$$
This is the unit <span efficiency factor> case.<Force Calculations for Lifting Line (16.100) by MIT OpenCourseWare>[pp.1–3, lift integral and elliptic loading]{ref} For stipulated $b=10\,\mathrm m$, $U=30\,\mathrm{m/s}$, $\rho=1.2\,\mathrm{kg/m^3}$ and $L=6000\,\mathrm N$, it gives $\Gamma_0\simeq21.22\,\mathrm{m^2/s}$, $w\simeq1.061\,\mathrm{m/s}$ and \b[$D_i\simeq212.2\,\mathrm N$].
A proposed test compares sectional <surface pressure integration for airfoil lift> with a traverse of the downstream <velocity> field. The two observations test the predicted loading and wake, rather than only one total force. <Flow separation>, strong sweep and <compressibility> require further modeling; the classical calculation does not predict all <drag> or <aerodynamic stall>.
Back to article page
